Download Stars: radius and mass

Stars
• Stellar radii
– Stefan-Boltzman law
• Measuring star masses
How to measure the size of star?
• If we know the distance to a star and can resolve its disk,
then we can use trigonometry.
• However, this has been successful for a very small
number of stars
How to measure the size of star?
• Light from single source will produce an interference pattern if passed
through two slits.
• The interference pattern for two sources, or one extended source, will be
smeared out. By accurately measuring the interference pattern caused
by combining the light from two telescopes, one can estimate the size of
the object or the separation of close binary stars.
• Need b > /, where  = wavelength,  = angular size.
Luminosity of a ‘Black Body’ Radiator
For the spherical object, the total power radiated
= the total luminosity is:
L=
4
2
4R T
T = temperature
 = Stephan-Boltzman constant
= 5.6710-8 W/m2 ·K4
R = radius
Luminosity of a ‘Black Body’ Radiator
Suppose the radius of the Sun increased by a
factor of 4 but the luminosity remained the
same. How would the surface temperature of
the Sun change?
L  4R T  4R T
2
1
4
1
1
2
2
2
1
2
4
2
 R1 
1
1
T2    T1    T1  T1
2
4
 R2 
Stars come in a variety of sizes
• If we know luminosity and temperature,
then we can find the radius:
L = 4R2T4
• Small stars will have low luminosities
unless they are very hot.
• Stars with low surface temperatures must
be very large in order to have large
luminosities.
Masses of stars
• Essentially all of the mass
measurements that we have for stars
are for stars in binary systems – two
stars orbiting each other.
• The mass of the stars can be measured
from the orbital period and either the
stellar velocities or the separation
between the stars.
Kepler’s 3rd Law applied to Binary Stars
4
3
2
a P
G (m1  m2 )
2
Where:
• G is gravitational constant
• G = 6.67·10-11 m3/kg-s2 in SI units
• m1, m2 are masses (kg)
• P is binary period (sec)
• a is semi-major axis of orbit (m)
Simplified form of Kepler’s 3rd law
using convenient units
3
a
M1  M 2  2
P
Where M in solar masses
a in AU
P in Earth years
Example: a = 0.05 AU, P = 1 day = 1/365 yr, M1 + M2 = 16.6 Msun
a is semi-major axis of orbit = half the length of yellow line
Binary star systems
• Double star – a pair of stars located at nearly the
same position in the night sky.
– Optical double stars – stars that appear close together,
but are not physically conected.
– Binary stars, or binaries – stars that are gravitationally
bound and orbit one another.
• Visual binaries – true binaries that can be observed
as two distinct stars
• How to distinguish true binary stars systems?
Visual Binary Star Krüger 60
(upper left hand corner)
About half of the stars visible in the night sky
are part of multiple-star systems.
Mizer-Alcor : A double-double-double system!
10 arcmin
Alcor
Mizar A
Mizar A+B
Mizar B
Note: Mizar B is
also a binary
with period of 6
months!
Mizar A
(Binary, P = 20.5 days)
Mizar observations using the NPOI
(Naval Prototype Optical Interferometer, near Flagstaff Arizona)
Determining masses of Mizar-A binary stars from
observations of period, angular separation, distance
1. Distance (from parallax) d = 25.4 pc
2. Max. angular separation (NPOI meas.)  = 0.0192"
3. Physical separation D = θ·d = 0.49 AU
4. Sum of masses (Kepler’s 3rd law)
a 3 (0.49 / 2)3
M1  M 2  2 
 4.7
2
P
0.056
5. Orbit shows a1 ~ a2 (NPOI meas.) so:
M 1a1  M 2 a2
M 1  M 2  2.3M 
What about unresolved binary systems?
• Spectroscopic binaries
– There are lines in the spectrum of almost every star.
– These lines will be Doppler shifted by the motion of the star in the
binary.
– The shifts for the two stars will be out of phase, one star is moving
towards us as the other is moving away.
– Can determine binary nature by looking for motions of lines (versus
wavelength) in spectra.
Determining component masses of eclipsing
binaries using velocity curves
1. Determine semi-major axis using
observed velocity (V), period (P)
2a1
v1 
P
a  a1  a2
2a2
v2 
P
2. Determine sum of masses using
Kepler’s 3rd law
3
a
M1  M 2  2
P
3. Determine mass ratio using a1, a2
M1a1  M 2 a2 or M1v1  M 2v2
4. Use sum, ratio to determine component masses
a1
a2
a = a 1 + a2
Tilt of Binary Orbits
We have been assuming that we see the binary system face on
when imaging the orbit and edge-on when measuring the velocity.
In general, the orbit is tilted relative to our line of sight. The tilt, or
inclination i, will affect the observed orbit trajectory and the
observed velocities. In general, one needs both the trajectory and
the velocity to completely determine the orbit or some
independent means of determining the inclination.
Light curves of eclipsing binaries provide
detailed information about the two stars.
Light curves of eclipsing binaries provide
detailed information about the two stars.
In general, need to simulate orbit (period and separation), sizes
and temperatures of stars, and heating to accurately reproduce the
orbital light curve. Often useful to obtain light curves in multiple
wavebands.